"I've heard you never know what you'll get as far as accuracy"
Pre marlin take over that did seem to be the case, New manufature does seem to be way more consistant ( that 22-250 is from after the takeover). there are a couple of things to look for that will help predict accuracy.
1. Take the forend off there should be no movement .
2. Since the forend if off smoke the latch pin shelf and reistall the barrel the mark in the soot should be even and about 1/8" deep
3. Take feeler guage and put it in on the breech face across the middle of the chamber. Ideally you should be able to close but not pull out .0015" you can go as high as .003" but that is real borderline. if you have bigger than that chances are you have problem 1.
My 243 had a .006" gap and it rattled to fix it I put a .0045 shim (Piece of feller guage), between the hindge pin and the notch. THat really tightened it up and the groups went under 1.5 immediatly.
There are other things, like the art/magic of how forearms and how the are fitted effect accuracy. You may want to go as far as slugging the barrel especially on an older handi to make sure it is not over-bore.
Realisitcally 1 - 1.5" @ 100 is the expected accuracy ( for bottle necked rounds) but many do much better, MOA or better. Two things when shooting groups with a handi.
1. Follow though on the trigger, if you don't the transfer bar can fall out of place causing inconsistant ignition.
2. Generally handi's like to be rested not on the forearm like a bolt gun but close to the hindge pin.Here is a picture of my homemade shooting rest to show you what I mean.
In my experience Handis can be more load sensitive than other guns. With my 204 with some loads +- .2 from the sweet spot and the groups opened up. With my 204 I changed my load develpment. I went max - 10% with intervals of .5 of a grain till max - 1 grain then up by .2. I fully believe that if I had done that with the 243 before I sold it it would have been pushing MOA maybe sub with a trigger job.